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Mirrors > Home > ILE Home > Th. List > exmidontri2or | Unicode version |
Description: Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.) |
Ref | Expression |
---|---|
exmidontri2or | EXMID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmidontriim 7202 | . . 3 EXMID | |
2 | onelss 4372 | . . . . . . . 8 | |
3 | 2 | adantl 275 | . . . . . . 7 |
4 | orc 707 | . . . . . . 7 | |
5 | 3, 4 | syl6 33 | . . . . . 6 |
6 | eqimss 3201 | . . . . . . . 8 | |
7 | 6, 4 | syl 14 | . . . . . . 7 |
8 | 7 | a1i 9 | . . . . . 6 |
9 | onelss 4372 | . . . . . . . 8 | |
10 | 9 | adantr 274 | . . . . . . 7 |
11 | olc 706 | . . . . . . 7 | |
12 | 10, 11 | syl6 33 | . . . . . 6 |
13 | 5, 8, 12 | 3jaod 1299 | . . . . 5 |
14 | 13 | ralimdva 2537 | . . . 4 |
15 | 14 | ralimia 2531 | . . 3 |
16 | 1, 15 | syl 14 | . 2 EXMID |
17 | ontri2orexmidim 4556 | . . . 4 DECID | |
18 | 17 | adantr 274 | . . 3 DECID |
19 | 18 | exmid1dc 4186 | . 2 EXMID |
20 | 16, 19 | impbii 125 | 1 EXMID |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 703 DECID wdc 829 w3o 972 wceq 1348 wcel 2141 wral 2448 wss 3121 c0 3414 csn 3583 EXMIDwem 4180 con0 4348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-uni 3797 df-tr 4088 df-exmid 4181 df-iord 4351 df-on 4353 df-suc 4356 |
This theorem is referenced by: onntri52 7221 |
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